In order to compute the one-loop amplitudes used to generate EFT data
we applied standard amplitudes methods, including spinor
helicity [1]}, generalized unitarity [2]}, [3]}, the double copy [4]}, [5]}, [6]}
and integration by parts [7]}, [8]}. Using
generalized unitarity we obtained all integral coefficients except for
a few whose integrals have no unitarity cuts in any channel. We fixed
the coefficients of the latter integrals by using the known
ultraviolet properties of the amplitude. To fully utilize this
information we made use of overcomplete integral basis that contains
higher-dimensional integrals, but whose coefficients do not depend on
the spacetime dimension or equivalently the dimensional-regularization parameter
\(\epsilon \) . In addition, we also demonstrated that these coefficients
do not depend on the mass \(m\)  of the particles circulating in the
loops. The existence of such a basis imposes constraints
that makes it reasonably straightforward to determine any pieces not captured by
the \(s\) , \(t\)  and \(u\)  channel unitarity cuts. In
this basis we used the fact that the coefficient of the potential
\(1/\epsilon \)  logarithmic ultraviolet divergence is
zero [9]} to completely fix the remaining integral
coefficients. A basis without \(\epsilon \) -dependent coefficients always
exists for one-loop problems [10]}, however the lack of
\(m\) -dependence appears to be special to our case of a closed massive loop
with external massless particles.
